A356460 Expansion of e.g.f. Product_{k>0} B(x^k)^k where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

1, 1, 6, 35, 303, 2772, 32903, 410335, 6051692, 95183187, 1675869175, 31437027030, 644157830077, 13976891765137, 325719071472590, 8007861177420275, 208953947981129027, 5725964099963426924, 165258064179632753563, 4987477844227598529047

Offset: 0

Author

Seiichi Manyama, Aug 08 2022

Keywords

• nonn

Crossrefs

Cf. A209902, A209903.

Cf. A000110, A354863, A356461.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1)^k)))

PARI

a354863(n) = n!*sumdiv(n, d, n/d/d!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354863(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

E.g.f.: Product_{k>0} exp(k * (exp(x^k)-1)).

a(0) = 1; a(n) = Sum_{k=1..n} A354863(k) * binomial(n-1,k-1) * a(n-k).